In many active control applications, cancellation is required only at discrete frequencies where tonal disturbances exist. While adaptive filter-based systems using the filtered-X and filtered-U LMS update methods are effective in canceling broadband disturbances, adaptive filters can be computationally burdensome when canceling tones, especially if the number of cancellation actuators and error sensors is large.
Moreover, in tonal applications, broadband adaptive filters can sometimes be over-parameterized. Over-parameterization can lead to lack of persistent excitation within the filter. Another problem with over-parameterization is that the potential exists for output from separate cancellation actuators to increase yet cancel one another, thus avoiding detection by error sensors. This can lead to unnecessarily high power consumption, and instability. These problems can be remedied via leakage methods, but most leakage methods can compromise performance.
Another characteristic of adaptive filters is that, with respect to adaptation, there exists an interdependence among all tap weight values in the filter. This interdependence can reduce convergence rate.
As an alternative to the filtered-X and filtered-U LMS updates, Fast Fourier Transforms can be used to transform signals from error sensors into the frequency domain as a set of complex numbers. The real and imaginary part can then be separately filtered through the complex transpose (i.e. Hermitian transpose) of a transfer function representing the speaker-error path. This procedure accounts for phase shifts and delays through the speaker-error path and can be used to improve stability. But, this procedure can be quite burdensome computationally, and also increases time to track changes in the system.
It is therefore desirable to provide a method of attenuating selected tones in a system input that avoids over-parameterization, accounts for phase shifts or delays through the speaker-error path, reduces the required amount of processing and improves the convergence rate of the system, while at the same time maintains long term stability.